Randomized low-rank approximation for symmetric indefinite matrice
The Nystr¨om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In this work, we first identify the main challenges in finding a...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2023
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| _version_ | 1826311323413118976 |
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| author | Taejun, P Nakatsukasa, YN |
| author_facet | Taejun, P Nakatsukasa, YN |
| author_sort | Taejun, P |
| collection | OXFORD |
| description | The Nystr¨om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In this work, we first identify the main challenges in finding a Nystr¨om approximation to symmetric indefinite matrices. We then prove the existence of a variant that overcomes the instability, and establish relative-error nuclear norm bounds of the resulting approximation that hold when the singular values decay rapidly. The analysis naturally leads to a practical algorithm, whose robustness is illustrated with experiments. |
| first_indexed | 2024-03-07T08:06:33Z |
| format | Journal article |
| id | oxford-uuid:6a4589b1-18f2-4d76-8791-db6de60b40da |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:06:33Z |
| publishDate | 2023 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:6a4589b1-18f2-4d76-8791-db6de60b40da2023-11-06T09:21:36ZRandomized low-rank approximation for symmetric indefinite matriceJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:6a4589b1-18f2-4d76-8791-db6de60b40daEnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2023Taejun, PNakatsukasa, YNThe Nystr¨om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In this work, we first identify the main challenges in finding a Nystr¨om approximation to symmetric indefinite matrices. We then prove the existence of a variant that overcomes the instability, and establish relative-error nuclear norm bounds of the resulting approximation that hold when the singular values decay rapidly. The analysis naturally leads to a practical algorithm, whose robustness is illustrated with experiments. |
| spellingShingle | Taejun, P Nakatsukasa, YN Randomized low-rank approximation for symmetric indefinite matrice |
| title | Randomized low-rank approximation for symmetric indefinite matrice |
| title_full | Randomized low-rank approximation for symmetric indefinite matrice |
| title_fullStr | Randomized low-rank approximation for symmetric indefinite matrice |
| title_full_unstemmed | Randomized low-rank approximation for symmetric indefinite matrice |
| title_short | Randomized low-rank approximation for symmetric indefinite matrice |
| title_sort | randomized low rank approximation for symmetric indefinite matrice |
| work_keys_str_mv | AT taejunp randomizedlowrankapproximationforsymmetricindefinitematrice AT nakatsukasayn randomizedlowrankapproximationforsymmetricindefinitematrice |